Does anyone sell windows that I can just put inbetween the studs with a
short section of 2x4 above and below to frame it in?

The studs are 14" on center.

I know they'd be small, but if I put a lot of them in, it seems like the
end result would be fine.

I can't think of any particular reason you couldn't mount solatubes
sideways,
except that they'd be ugly. And yes, you can have made windows that
will fit between the studs.

But it's not obvious to me that, once you've punched a hole in the
drywall and siding anyway, it's particularly more of a PITA to stick in
a header and double up studs. Especially if you have to put in more
than one little window to make up the difference.